A system of linear equations is partition regular if, whenever the natural numbers are finitely coloured, the system of equations has a monochromatic solution. Partition regularity can also be defined over the rationals, and if the system of equations is finite then these notions coincide. We construct an example of an infinite system which is partition regular over the rationals but not the naturals. The proof is based on examining what happens when you take iterated sumsets and difference sets of subsets of the integers with positive upper density.

Take a long (proportional to n^2) random walk W in a quasirandom graph G, then delete all the edges traversed by W. Must the new graph G' be quasirandom? We'd like to say yes, for the following reason: W visits every vertex about the same number of times, so we lose the same number of random edges at every vertex. In the case where the minimum degree of G is large, this argument is essentially correct. If G has some vertices of very low degree then it breaks down because the random walk can get stuck in clusters of low degree vertices. However, a more sophisticated argument can recover a result that is almost as strong.

The proofs both fall into two parts: first show that the random walk does not differ too much from a process that has much more independence, then exploit that independence by applying standard concentration results to show that things work with high probability. It turns out that our results can be tweaked to apply to the more general case of random homomorphisms of trees (rather than paths) provided the maximum degree of the tree isn't too large, so we indicate the necessary changes at the end of the paper.

Borg asked what happens to the Erdős-Ko-Rado theorem if we only count sets meeting some fixed set X, and answered the question for |X| ≥ r, the size of the sets in the set family. This paper answers the question for |X| < r, provided n, the size of the ground set, is sufficiently large.

How large can an independent set in the discrete cube be if it contains equal numbers of sets of even and odd size? Take odd sets starting from the bottom of the cube, and even sets starting from the top. Proving that this works uses an isoperimetric inequality: if you know the proof of Harper's theorem that uses codimension 1 compressions then you know how to prove the inequality that's quoted without proof in this paper.